Abstract

We demonstrate single mode terahertz (THz) guidance in hollow-core kagome microstructured fibers over a broad frequency bandwidth. The fibers are characterized using a THz time-domain spectroscopy (THz-TDS) setup, incorporating specially designed THz lenses to achieve good mode overlap with the fundamental mode field distribution. Losses 20 times lower than the losses of the fiber material are observed in the experiments, as well as broad frequency ranges of low dispersion, characteristic of hollow-core fibers.

1. Introduction

A highly dynamic research field, THz radiation has inspired many innovations and applications that transcend its boundaries. Recent interest in THz radiation has led to the development of state-of-the-art THz generation and detection systems. These include coherent, wideband THz DC biased emitters and electro-optic sampling [1

], where the longer wavelengths (compared to the visible spectrum) make experimental investigation somewhat easier.

Whilst free-space propagation of THz has been widely used, confining the radiation to waveguides has been challenging due to a lack of highly transparent materials. THz waveguides composed of a variety of dielectric and metallic waveguides have been demonstrated over the years. Reported examples include unclad sapphire-fibers [9

], also called photonic crystal fibers (PCF), to THz waveguiding, as these offer possibilities for low loss and also flexible control over the guided mode characteristics such as dispersion. This control is achieved through the microstructured geometry of the waveguide which confines the radiation through holes that extend the length of the waveguide; this has been studied extensively in the context of visible and infrared light. In recent years, several different implementations of MOF for THz waveguiding using polymers have shown attractive results with a lot of potential. Waveguides using a hexagonal array of holes around a solid core, typical of single-mode MOF designs, have been demonstrated using HDPE [17

]. These had core sizes between 0.5 and 1 mm and achieved 0.5 cm−1 for 0.2-3.0 THz, 0.3 cm−1 at 1 THz and 0.09 cm−1 for 0.2-1.5 THz, respectively. Similarly, a Zeonex® single-mode suspended-core design – a solid core surrounded by five air holes – with a 0.4 mm core, was reported with an average loss of 0.13 cm−1 across 0.3-0.9 THz [20

]. In all but the last example, the THz radiation is guided in a high index core via modified total internal reflection, and hence is subject to the material absorption losses, while the PMMA ring-structured Bragg fiber is an example of THz guidance in a low index (hollow) core dielectric waveguide by means of a photonic bandgap. The hollow core means that material absorption can be almost neglected due to the small overlap of the propagating radiation with the waveguide material.

In this work, we investigate the propagation of THz radiation in the hollow core of kagome microstructured fibers. Other types of hollow-core fibers confine the mode in the core by means of a photonic bandgap in the cladding – the cladding does not support propagating modes for a particular range of wavelengths and mode effective indices (the bandgap). Radiation can be confined to the core by the bandgap as the absence of cladding modes prevents it from escaping the core through coupling to the cladding. In contrast, guidance in kagome fibers is due to the inhibited coupling mechanism; the cladding does not support bandgaps. Power in the core mode may couple to cladding modes when the two are resonant, and this is most pronounced at the cut-off frequencies of the cladding modes, resulting in high transmission loss. Away from these cut-off frequencies, although modes in the cladding exist (as there is no bandgap) the radiation in the core is inhibited from coupling to them and thus remains propagating in the core [22

]. This inhibited coupling from the core to the cladding is the result of a low density of cladding modes, a small physical overlap between these and the core mode, and a mismatch in their spatial frequencies [22

We report on the fabrication of these fibers and the characterization of their spectral transmission, modal distribution, loss and dispersion using time domain spectroscopy (TDS), and compare the experimentally determined transmission to simulations.

2. Fiber fabrication and experimental method

Fig. 1 (a) Optical micrograph of PMMA kagome fiber showing the core and the innermost 4 of the 6 rings of the cladding, (b) THz symmetric-pass lens with a diameter of 50 mm. (c) THz-TDS setup used to characterize the microstructured fiber (in magenta color). AOM stands for acousto-optic modulator.

we show a micrograph of the PMMA (n = 1.60, α = 12 cm−1 @ 1 THz for the bulk material) kagome microstructured fiber. The preform was made by stacking PMMA tubes (3 mm inner diameter/5 mm outer diameter) in a triangular lattice that gives the appearance of a kagome (‘eye of the basket’ in Japanese) structure. The core was formed by removing 7 tubes from the stack, leaving 6 hexagonal rings of tubes to form the cladding. The preform was stretched from 80 mm outer diameter directly to the desired outer diameters: 5 mm and 6.8 mm. These correspond to kagome fibers with core diameters of 1.6 mm and 2.2 mm respectively. For future reference, we will refer to the former as Kagome-1 and to the latter as Kagome-2.

To obtain a good mode matching, we employed specially designed symmetric pass lenses (Fig. 1 (b)); the lens design is discussed in Ref. [28

]. We choose the design parameters of the lens to achieve a focal length of 75 mm and a numerical aperture of 0.33. The lenses are made of ultrahigh molecular weight polyethylene (UHMWPE). The details of the THz-TDS setup (Fig. 1 (c)) are discussed in Section 3 of [20

3. Results and discussion

3.1 Temporal evolution of the guided THz wave

Fig. 2 (a) Temporal signals as measured by the lock-in amplifier for the reference pulse and 30 mm of Kagome-1 and Kagome-2 fibers. (b) Spectrograms of the reference, Kagome-1 and Kagome-2 on a logarithmic scale. The amplitudes are normalized with respect to the peak value of the reference signal. Horizontal white dashed lines on Kagome-1 and Kagome-2 spectrograms indicate the high loss frequency region associated with the cladding modes cutoffs obtained from simulation results.

the temporal signals obtained in the experiments. We note that there is very little time delay introduced to the setup when the fibers are inserted between the two lenses, indicating that the effective core index is very close to 1, as expected for an air core. Several fiber lengths were measured; six pieces of fiber with length ranging from 15 mm up to 45 mm for Kagome-1 and five pieces of fiber with lengths from 20 mm up to 45 mm for Kagome-2. Using the measured temporal scans, the respective short-time Fourier transform plots were computed, and these are shown in Fig. 2 (b). The plots show the frequency-time spectral density of the signals, normalized to the peak value of the reference signal density, on a logarithmic scale. Quantitative observations can be made from these spectrograms. Firstly, one notices that the transmission bands for both fibers arrive almost at the same time as the reference for all frequencies, i.e. there is little group delay in the modes that are guided in the core of these fibers and similarly low dispersion. Secondly, we note that the transmission window is widened in the fiber with the larger core diameter (Kagome-2).

3.2 Characteristics of the fundamental mode

Figure 2 (b) also shows the calculated frequency cut-offs (white dashed lines) for certain modes of the cladding struts, occurring at 0.7 THz for Kagome-1 and 0.6 THz for Kagome-2. Following the outline of the inhibited coupling guidance mechanism, the fibers are expected to have higher transmission losses at these frequencies (as well as at other cut-offs, which occur outside the frequency range investigated). These expected loss peaks are apparent in Figs. 3 (a) and (b)

Fig. 3 (a), (b) The attenuation constant for Kagome-1 and Kagome-2 fibers, respectively. The solid red lines indicate the simulation results obtained from MODE and the blue-circles the experimentally obtained data. The grey area indicates the region where a high loss in the propagating mode is expected from MODE, and the light-cyan area corresponds to a low signal-to-noise ratio. (c), (d) The field distribution as observed experimentally in the 30 mm lengths of the fibers. The peak frequencies correspond to the highest transmission amplitudes recorded whilst the resonant frequencies correspond to those at which a higher loss is expected due to the mode leakage into the cladding material, as predicted using the MODE simulation results.

(highlighted grey area) which shows the loss spectra of the two samples, obtained by normalizing their transmission with respect to the reference scan. The data presented in Figs. 3 (a), (b) and the fibers coupling efficiencies are calculated from comparisons of the ratios of the spectrum obtained from the measured signals of the fibers with various lengths.

Away from the loss peaks these fibers show low-loss guidance through the hollow core. For Kagome-1, the average power attenuation coefficient is about 1 cm−1 over 0.75-1.0 THz. For the larger core Kagome-2 the average power attenuation coefficient is about 0.6 cm−1 over 0.65-1.0 THz. We note that within this transmission window, the propagation loss stays well below 1 cm−1, a value achieved by some of the conventional THz waveguides noted above. Comparing these values to the attenuation of the PMMA material itself (12 cm−1 @ 1 THz), the loss shows a marked decrease, on average 20 times lower than the material for Kagome-2, as expected from confining the radiation to a hollow core. Comparing the two fibers with each other, we note that Kagome-2 has a loss 40% lower compared to Kagome-1. This is justified as the field strength of the guided mode is still quite appreciable at the core-cladding interface for Kagome-1, and this spatial overlap with the lossy dielectric walls adds to propagation losses [22

]. The extent of this spatial overlap is reduced in Kagome-2 as the struts surrounding the core are farther from the center of the core and the field is less intense at the core boundary.

In the left panels of Figs. 3 (c) and (d), we verify experimentally that only the fundamental mode propagates through both of the kagome fibers despite the increase in core diameter by about 38%. The images are taken by performing a grid scan at the output end of the fiber. At the peak frequencies (chosen where the highest transmission amplitudes are obtained), we observe strong confinement of the fields in the core region. The observed experimental data show good agreement with the simulated distribution of the fundamental mode. On the right panels of Figs. 3 (c) and (d), the field distribution for each fiber is shown at one of the high loss frequencies. For the fiber lengths used in the experiments, the fundamental mode did not totally vanish at these high loss frequencies but the power leaked into the cladding is detected. This coupling into the cladding causes higher losses for two reasons: firstly the attenuation in the cladding structure is higher due to the bulk absorption coefficient of PMMA, and secondly the cladding modes are also coupled to radiation modes in the surrounding air, and therefore offer a path to escape the waveguide structure entirely.

3.3 Comparison with simulation results

The experimental measurements of Fig. 3 show good agreement with the simulation results obtained from MODE [29

]. The fully vectorial approach using a finite difference frequency domain (FDFD) solver employed by this commercially available program takes the image of the cross-section of the fiber and the material parameters (refractive index and attenuation coefficient) as inputs to compute the results shown here. Discrepancies between the simulation results and the measured loss spectra are attributed to small but non-negligible non-uniformity along the lengths of the fibers. The peaks in the loss spectra are susceptible to changes in strut thickness as the frequency cutoffs of the cladding modes depend on their size. Additionally, we considered a structure identical to Kagome-2 but with a reduced material absorption compared to PMMA of α = 0.2 cm−1 @ 1 THz for the bulk material (this is the absorption coefficient for Zeonex® [20

], one of the lowest reported values for a polymer). Simulations of such a fiber yield that the calculated losses for the guided mode are about the same value as that of obtained for the real PMMA fiber at 1 THz. This indicates that the losses of the guided mode for this structure are not greatly influenced by the bulk material losses, and is consistent with what was stipulated in the comparison above between the loss of Kagome-1 and −2, i.e. that the core of Kagome-2 is sufficiently large to effectively eliminate material loss. Therefore the fiber material choice is not a critical factor in such cases.

3.4 Effective mode index and group velocity dispersion

The phase of the different frequency components was extracted from the Fourier transforms of the measured temporal signals and un-wrapped to take into account the 2π phase jumps that can occur between the two consecutive frequency points. The corrected phase details were used to calculate the mode effective index of the propagating fundamental mode, as shown in Figs. 4 (a) and (b)

Fig. 4 (a), (b) The effective phase index and (c), (d) the group velocity dispersion parameters, β2, for Kagome-1and Kagome-2 fibers, respectively. In all subfigures, the solid red lines indicate the simulation results obtained from MODE and the blue-circled lines are the experimentally obtained data. The grey area indicates the region where a high loss in the propagating mode is expected from MODE, and the light-cyan area is the region where a low signal-to-noise ratio results in more noisy data.

. The agreement between the experimental data and the simulations is very good for Kagome-1, and still good for Kagome-2. The oscillations in the mode’s effective index around the high loss region (highlighted grey area in Figs. 4 (a) and (b)) arise from avoided crossings between the core mode and modes supported by the cladding. This is most pronounced at the edge of the transmission window approaching the high loss band, as is typical for hollow-core fibers.

We extend our analysis to the group velocity dispersion (GVD) represented by the β2(β2=∂2β/∂ω2)parameter, where ω is the angular frequency and β is the frequency-dependent propagation constant defined asβ=nω/c,nbeing the effective mode index. Figures 4 (c) and (d) show thatβ2stays relatively small in the high-transmission windows, typical of hollow-core fibers. In the regimes where high signals are measured, the β2 data determined from the experiments follow the pattern of the simulated values. The larger fluctuations in the experimental data arise from the amplification of noise that occurs when a function is differentiated twice. This effect is more noticeable at higher frequencies due to the lower signal-to-noise ratio. The overall low GVD values for both fibers confirm the picture already indicated by the spectrograms (Fig. 2 (b)): For all three scans (reference and the two fibers) the mean time for each frequency component is about the same, i.e. all components travel with about the same speed. The low GVD values exhibited by these fibers are comparable to that of the experimentally achieved low dispersion waveguides as reported in [13

]. Overall, the simulations show that both Kagome-1 and Kagome-2 fibers can achieve very low dispersion over a wide frequency range.

Finally, we further established that the guided radiation through the fibers is well confined within the fiber structure, although this is already clear from Figs. 3 (c) and (d). No change in measured losses and dispersion was observed when the fiber mounting was changed from a thin paper aperture to a metal disc aperture. In terms of handling, the fibers are robust and their structural integrity is preserved under modest pressure. Therefore, they do not require careful handling, an advantage compared to e.g. porous fibers [30

]. Additionally, the use of the specially designed symmetric-pass lenses ensures good mode overlap between the input field and the fundamental mode of the fibers. Calculations of the coupling efficiency from our measurement data indicate that values as high as 60% are achieved for both fibers.

4. Conclusions and future work

We demonstrated wideband, low loss propagation of THz radiation in hollow-core kagome fibers. The fibers investigated have core diameters at least 6 times larger than propagating wavelengths, yet we experimentally observe propagation of the fundamental mode only. These relative core sizes are comparable to what is observed for single-mode solid core waveguides, but are much larger than single-mode hollow metallic waveguides. The properties of these kagome fibers will be advantageous for many single mode applications that require larger core diameters. The measured attenuation coefficients are approaching those of more conventional THz waveguides. The average losses measured from these microstructured fibers are below 1 cm−1 in the region from 0.75 to 1.0 THz for Kagome-1, and about 0.6 cm−1 in the region from 0.65 to 1.0 THz for the Kagome-2 fiber. The propagation losses show a reduction of about 20 times over the material loss of the PMMA itself.

All experimental results agree well with the MODE simulations, and the low-GVD properties of the fibers are confirmed. We also simulated the effect of a lower bulk material loss for the fiber and found that the simulated losses of the guided mode in the air core are almost the same value in the case of the larger core diameter. Consequently, this shows that the choice of the fiber material is not a critical factor in approaching low-loss guidance in a hollow-core MOF of sufficiently large cores. Due to the good mode overlap between the input mode and the fundamental mode of both kagome fibers, coupling efficiencies as high as 60% were achieved. We believe that such hollow-core waveguides are promising for the transmission of THz radiation, as they can potentially achieve lower losses and low dispersion over large frequency bands than is possible, for example, in dielectric-based solid-core waveguides. The low dispersion properties may be particularly significant, given that common THz systems such as THz-TDS use pulsed sources.

In the light of these results, we believe that the next step forward is to minimize the overall outer diameter of the fibers for more compact, practical THz fibers. This can be done by eliminating the outermost cladding layers – this will reduce the outer diameter of the fibers without impacting their properties as waveguides; such a simplified structure has been theoretically discussed in [31

]. In reducing the outer diameter of the kagome fiber, it allows the possibility of guiding THz radiation in a more flexible waveguide.

Hollow-core waveguides can also be an efficient means for directional delivery of THz radiation as they can replace off-axis parabolic mirrors that are vulnerable to aberrations. The direct coupling of THz pulses from, for example, a photoconductive antenna emitter can be efficient for larger core waveguides. Kagome fibers are a good candidate for this purpose as they offer THz transmission over a wide frequency bandwidth as well as keeping the dispersion of the travelling pulse as low as possible. Further studies on the optimal waveguide parameters for this task are needed to address the issues such as single mode operation limits.

Acknowledgments

J. A. acknowledges the financial support of Industrial Research Limited (IRL) New Zealand, and thanks David Wu for his input. A. A. is supported by an Australian Research Council Australian Research Fellowship and thanks B. Kuhlmey for his input. S. L.-S. is supported by an Australian Research Council Australian Postdoctoral Fellowship. This work was performed in part at the OptoFab node of the Australian National Fabrication Facility, a company established under the National Collaborative Research Infrastructure Strategy to provide nanofabrication and microfabrication facilities for Australian researchers.

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